How Many Kähler Metrics Has a K-3 Surface?

Todorov, Andrei N.

doi:10.1007/978-1-4757-9286-7_18

Cited by 6 publications

(1 citation statement)

References 5 publications

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“…The infinitesimal deformations (meaning in this situation the h ij ∈ N such that ∆ L h ij = 0) of Kähler Ricci-flat metrics on K3 manifolds are known to actually correspond to Ricci-flat metrics [25,23]. Once again, these metrics form a submanifold E of Ricci-flat fixed points.…”

Section: K3 Manifoldsmentioning

confidence: 99%

A gradient flow for worldsheet nonlinear sigma models

2006

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We discuss certain recent mathematical advances, mainly due to Perelman, in the theory of Ricci flows and their relevance for renormalization group (RG) flows. We consider nonlinear sigma models with closed target manifolds supporting a Riemannian metric, dilaton, and 2-form B-field. By generalizing recent mathematical results to incorporate the B-field and by decoupling the dilaton, we are able to describe the 1-loop β-functions of the metric and B-field as the components of the gradient of a potential functional on the space of coupling constants. We emphasize a special choice of diffeomorphism gauge generated by the lowest eigenfunction of a certain Schrödinger operator whose potential and kinetic terms evolve along the flow. With this choice, the potential functional is the corresponding lowest eigenvalue, and gives the order α ′ correction to the Weyl anomaly at fixed points of (g(t), B(t)). The lowest eigenvalue is monotonic along the flow, and since it reproduces the Weyl anomaly at fixed points, it accords with the c-theorem for flows that remain always in the first-order regime. We compute the Hessian of the lowest eigenvalue functional and use it to discuss the linear stability of points where the 1-loop β-functions vanish, such as flat tori and K3 manifolds.

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Section: K3 Manifoldsmentioning

confidence: 99%

A gradient flow for worldsheet nonlinear sigma models

2006

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SUSY Strings at Strong Coupling

Cecotti

2023

Theoretical and Mathematical Physics

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The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24

Taormina

Wendland

2013

J. High Energ. Phys.

114

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In view of a potential interpretation of the role of the Mathieu group M 24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different K3 surfaces to larger 'overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type A 24 1 , which is simultaneously compatible with the finite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice serves to express all these symplectic automorphisms as elements of the Mathieu group M 24 , generating the 'overarching finite symmetry group' (Z 2 ) 4 ⋊ A 7 of Kummer surfaces. This group has order 40320, thus surpassing the size of the largest finite symplectic automorphism group of a K3 surface by orders of magnitude. For every Kummer surface this group contains the group of symplectic automorphisms leaving the Kähler class invariant which is induced from the underlying torus. Our results are in line with the existence proofs of Mukai and Kondo, that finite groups of symplectic automorphisms of K3 are subgroups of one of eleven subgroups of M 23 , and we extend their techniques of lattice embeddings for all Kummer surfaces with Kähler class induced from the underlying torus.

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How Many Kähler Metrics Has a K-3 Surface?

Cited by 6 publications

References 5 publications

A gradient flow for worldsheet nonlinear sigma models

A gradient flow for worldsheet nonlinear sigma models

SUSY Strings at Strong Coupling

The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24

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